Coherent groups

Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups.

Instead of trying to motivate the study of coherent groups in my own words, let me instead just refer to Section 2 of the article `An Invitation to Coherent Groups‘ of Daniel Wise (DOI: 10.1515/9780691185897-014).

In this post I just wanted to quickly report on the following new result of Hughes-Kielak-Kropholler-Leary on this topic (arXiv:2302.03346).

It is motivated by an earlier result of Bieri-Strebel who found all the coherent groups in the class of (finitely generated) soluble groups: These are exactly the polycyclic groups, resp. a properly ascending HNN extension with a polycyclic vertex group.

The new result generalizes this to the class of (finitely generated) elementary amenable groups: These are exactly the virtually polycyclic groups, resp. a properly ascending HNN extension with a virtually polycyclic vertex group. So basically, their result states that if we look around in the more general class of all (finitely generated) elementary amenable groups, then we (virtually) don’t find anything more coherent there.

As a corollary of their result they conclude: Every coherent, finitely generated, elementary amenable group is virtually soluble and of type VF. The latter means that G contains a finite index subgroup H admitting a finite model for its Eilenberg-MacLane space K(H,1).

If you became interested in coherent groups, the above cited article by Daniel Wise seems to be a very good starting point to the theory. And, it contains a gazillion of open problems. For example, is SL(3,Z) coherent or not?